Perfect matchings in bipartite graphs have been extensively studied. If two graphs are isomorphic, then their perfect matchings are in a natural bijection. But what about the converse? Of course, it is not true, but what is true? Perfect matchings are also connected with the polytope of doubly stochastic matrices and its faces (a face can be viewed as a bipartite graph). This gives a geometric structure to the perfect matchings of bipartite graphs.